Finite Group Databasebeta

← #27320519

Subgroups — #27320519

Order 13568.

1802 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133176C211yesmaximal, characteristic
2C2×#16810504C212yesmaximal, complemented
3#20272979C211yesmaximal, characteristic, complemented
4#20272971C211yesmaximal, characteristic, complemented
5#20272968C211yesmaximal, characteristic, complemented
6#20272968C211yesmaximal, characteristic, complemented
7#20272964C212yesmaximal
8#20272958C212yesmaximal
9#20272939C212yesmaximal, complemented
10#20272938C211yesmaximal, characteristic
11C2×C53×#385C211yesmaximal, characteristic
12C2×#13133176—28no—
13C2×#13133176—24no—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919C2212yes—
17C22×C22⋊Dic53C2212yes—
18C22×C22⋊Dic53C2212yes—
19C22×C4×Dic53C2212yes—
20#16810504—28no—
21C2×#13133124C2212yes—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C24×C4×C53C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2211yescharacteristic
36C53×#385—24no—
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C23×Dic53C2312yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43#13133176—232no—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—24no—
47C2×C4⋊Dic53—24no—
48C2×#11719919—24no—
49C2×#11719919—24no—
50C2×#11719919—24no—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53—24no—
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—24no—
55C2×C22⋊Dic53—24no—
56C2×C22⋊Dic53—42no—
57C2×C4×Dic53—24no—
58C2×C4×Dic53—24no—
59C2×C4×Dic53D418yes—
60#13133124—24no—
61#13133124—22no—
62#13133124—24no—
63#13133124—22no—
64#13133124—24no—
65#13133124—24no—
66#13133124—24no—
67#13133124—24no—
68#13133124—42no—
69#13133124—42no—
70C25×C53C2311yescharacteristic
71C23×C4×C53—24no—
72C23×C4×C53—24no—
73C23×C4×C53—22no—
74C23×C4×C53—22no—
75C23×C4×C53D414yes—
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2311yescharacteristic
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C2×C4⋊C4×C53—24no—
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53—22no—
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53—22no—
89C2×C22⋊C4×C53—24no—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—22no—
92#95×C53—22no—
93#95×C53—22no—
94C24×C53—22no—
95C24×C53—21no—
96C24×C53—21no—
97C24×C53—21no—
98C24×C53—21no—
99C24×C53—22no—
100C24×C53—21no—
101C24×C53—22no—
102C24×C53—21no—
103C24×C53C2×D412yes—
104C24×C53#4114yes—
105C24×C53C2411yescharacteristic
106C22×C4×C53—216no—
107C22×C4×C53—216no—
108C22×C4×C53—28no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—28no—
115C22×C4×C53—28no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—22no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D414yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53#4112yes—
140C22×C4×C53#4112yes—
141C22×C4×C53#4112yes—
142C22×C4×C53#4112yes—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53C2×D414yes—
169C22×Dic53C2×D414yes—
170C22×Dic53#4114yes—
171C4⋊C4×C53—44no—
172C4⋊C4×C53—44no—
173C22⋊C4×C53—28no—
174C22⋊C4×C53—44no—
175C22⋊Dic53—216no—
176C22⋊Dic53—48no—
177C4⋊Dic53—28no—
178C4⋊Dic53—28no—
179#11719919—216no—
180C4×Dic53—216no—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53—24no—
184C23×C53—24no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—24no—
190C23×C53—24no—
191C23×C53—24no—
192C23×C53—24no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—42no—
203C23×C53#12314yes—
204C23×C53C22×D412yes—
205C23×C53C22×D411yescharacteristic
206C23×C53C2×#4112yes—
207C23×C53C2×#4112yes—
208C23×C53C2×#4111yescharacteristic
209C23×C53C2×#4111yescharacteristic
210C23×C53C2×#4111yescharacteristic
211C23×C53C2×#4111yescharacteristic
212C23×C53H(5,𝔽2)11yescharacteristic
213C23×C532−1+411yescharacteristic
214C23×C532−1+412yes—
215C2×C4×C53—232no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—42no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53#12014yes—
247C2×C4×C53#12314yes—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276#15114—531nomaximal, Sylow
277C22×C53—24no—
278C22×C53—24no—
279C22×C53—24no—
280C22×C53—44no—
281C22×C53—44no—
282C22×C53—44no—
283C22×C53—44no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53C2×#12011yescharacteristic
289C22×C53C2×#12311yescharacteristic
290C22×C53C2×#12311yescharacteristic
291C22×C53#52512yes—
292C22×C53#52512yes—
293C22×C53#52512yes—
294C22×C53#52511yescharacteristic
295C22×C53#53211yescharacteristic
296C22×C53#53212yes—
297C22×C53#53612yes—
298C22×C53#53611yescharacteristic
299C22×C53#53711yescharacteristic
300C22×C53#53712yes—
301C22×C53#55112yes—
302C22×C53#55212yes—
303C22×C53#55212yes—
304C22×C53#55212yes—
305C22×C53#55212yes—
306C22×C53#55211yescharacteristic
307C22×C53#55311yescharacteristic
308C22×C53#55511yescharacteristic
309C22×C53#55611yescharacteristic
310C22×C53#55612yes—
311C4×C53—24no—
312C4×C53—24no—
313C4×C53—42no—
314C4×C53—42no—
315C4×C53—81no—
316C4×C53—81no—
317Dic53—44no—
318Dic53—44no—
319Dic53—44no—
320Dic53—82no—
321C22×#117—531no—
322C2×#387—532no—
323C2×#385D5311yescharacteristic
324#1444—531no—
325#1442—531no—
326#1441—531no—
327#1441—531no—
328#1440—532no—
329#1439—532no—
330#1434—532no—
331C22⋊#95—531no—
332C2×C53—44no—
333C2×C53#238811yescharacteristic
334C2×C53#248411yescharacteristic
335C2×C53#248511yescharacteristic
336C2×C53#249811yescharacteristic
337C2×C53#249911yescharacteristic
338C2×C53#259912yes—
339C2×C53#260112yes—
340C2×C53#260912yes—
341C2×C53#261112yes—
342C2×C53#261611yescharacteristic
343C2×C53#284911yescharacteristic
344C24×C4C2×D5311yescharacteristic
345C2×#117—1068no—
346C2×#117—1064no—
347C22×C4⋊C4—532no—
348C22×C4⋊C4—531no—
349C22×C4⋊C4—531no—
350C22×C4⋊C4C2×D5311yescharacteristic
351C22×C4⋊C4C2×D5311yescharacteristic
352C22×C22⋊C4—532no—
353C22×C22⋊C4—532no—
354C22×C22⋊C4C2×D5311yescharacteristic
355C22×C22⋊C4C2×D5311yescharacteristic
356C22×C42—532no—
357#387—1068no—
358#385—24no—
359C2×#95—532no—
360C2×#95—531no—
361C2×#95—531no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95—532no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95—532no—
368C2×#95—532no—
369C2×#95C2×D5311yescharacteristic
370C2×#95C2×D5311yescharacteristic
371C53#1511411yescharacteristic, complemented, Sylow
372C25C22×D5311yescharacteristic
373C23×C4—24no—
374C23×C4—24no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—532no—
378C23×C4—532no—
379C23×C4—532no—
380C23×C4—532no—
381C23×C4C22⋊D5314yes—
382C23×C4C22×D5311yescharacteristic
383C23×C4C22×D5311yescharacteristic
384C23×C4C22×D5311yescharacteristic
385C23×C4C22×D5311yescharacteristic
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388#117—10632no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—1064no—
395C2×C4⋊C4—1064no—
396C2×C4⋊C4—1064no—
397C2×C4⋊C4—1064no—
398C2×C4⋊C4—1064no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1062no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—1064no—
407C2×C22⋊C4—1064no—
408C2×C22⋊C4—1064no—
409C2×C22⋊C4—1064no—
410C2×C22⋊C4—1064no—
411C2×C22⋊C4—2122no—
412C2×C42—538no—
413C2×C42—1064no—
414C2×C42—1064no—
415#95—22no—
416#95—22no—
417#95—1064no—
418#95—1064no—
419#95—1064no—
420#95—1064no—
421#95—1064no—
422#95—1064no—
423#95—1062no—
424#95—1062no—
425#95—2122no—
426#95—2122no—
427C24—22no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—21no—
432C24—21no—
433C24—21no—
434C24—21no—
435C24—22no—
436C24#1171994714yes—
437C24C2×C22⋊D5312yes—
438C24C23×D5311yescharacteristic
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—534no—
470C22×C4—534no—
471C22×C4—534no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4#1171994512yes—
498C22×C4#1171994512yes—
499C22×C4D4×D5314yes—
500C22×C4#1171994712yes—
501C22×C4#1171994712yes—
502C22×C4C2×C22⋊D5312yes—
503C22×C4C2×C22⋊D5312yes—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—1068no—
507C4⋊C4—1068no—
508C4⋊C4—10616no—
509C22⋊C4—28no—
510C22⋊C4—44no—
511C22⋊C4—10616no—
512C22⋊C4—2128no—
513C42—10616no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23#1313319814yes—
537C23C2×D4×D5312yes—
538C23C22×C22⋊D5311yescharacteristic
539C23#41×D5312yes—
540C23C2×#1171994712yes—
541C23C2×#1171994711yescharacteristic
542C23C2×#1171994711yescharacteristic
543C23C2×#1171994511yescharacteristic
544C23C2×#1171994511yescharacteristic
545C23#1313326311yescharacteristic
546C23#1313326411yescharacteristic
547C23#1313326512yes—
548C2×C4—232no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—1064no—
584C2×C4—1064no—
585C2×C4—1064no—
586C2×C4—1064no—
587C2×C4—1064no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4#1313318514yes—
608C2×C4#1313319814yes—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22C2×#1313318511yescharacteristic
621C22C2×#1313319811yescharacteristic
622C22C2×#1313319811yescharacteristic
623C22#1681112912yes—
624C22#1681112912yes—
625C22#1681113212yes—
626C22#1681113411yescharacteristic
627C22#1681116911yescharacteristic
628C22#1681117012yes—
629C22#1681119612yes—
630C22#1681119711yescharacteristic
631C22#1681120111yescharacteristic
632C22#1681120312yes—
633C22#1681129612yes—
634C22#1681130712yes—
635C22#1681130812yes—
636C22#1681130812yes—
637C22#1681131012yes—
638C22#1681131311yescharacteristic
639C22#1681131911yescharacteristic
640C22#1681133811yescharacteristic
641C22#1681134311yescharacteristic
642C22#1681134412yes—
643C4—24no—
644C4—24no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—4242no—
650C4—2124no—
651C4—2124no—
652C4—2124no—
653C2—44no—
654C2#2027793011yescharacteristic
655C2#2027900211yescharacteristic
656C2#2027901211yescharacteristic
657C2#2027916111yescharacteristic
658C2#2027916911yescharacteristic
659C2#2028028812yes—
660C2#2028031612yes—
661C2#2028039112yes—
662C2#2028040912yes—
663C2#2028044511yescharacteristic
664C2#2028300511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC2×C53×#385C2
prime radicalC25×C53C23
hypercenterC2×#385D53
hypocenterC53#15114
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#385

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16810504C2semidirect
3. #20272979C2semidirect
4. #20272971C2semidirect
5. #20272968C2semidirect
6. #20272968C2semidirect
9. #20272939C2semidirect
371. C53#15114semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#131331762
C2×#168105042
#202729792
#202729712
#202729682
#202729642
#202729582
#202729392
#202729382
C2×C53×#3852
#1511453